Panki Kim

We consider the fractional Laplacian -(-Δ) α/2 on an open subset in R d with zero exterior condition.We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

In this paper, we consider a large class of purely discontinuous rotationally symmetric Lévy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Lévy process. When D is a C 1 , 1 open set and the Lévy exponent of the process is given by Ψ ( ξ ) = ϕ ( | ξ | 2 ) with ϕ being a

The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.

Let μ = μ 1 ⋯ μ d be such that each μ i is a signed measure on \R d belonging to the Kato class \K d , 1 . The existence and uniqueness of a continuous Markov process X on \R d , called a Brownian motion with drift μ , was recently established by Bass and Chen. In this paper we study the potential theory of X . We show that X has a continuous density q μ and that there exist positive constants c i , i = 1 , ⋯ , 9 , such that c 1 e - c 2 t t - d 2 e - c 3 x - y 2 2 t ≤ q μ t x y ≤ c 4 e c 5 t t -

A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is−ϕ(−Δ), where ϕ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with ϕ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, r

A subordinate Brownian motion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent <inline-formula content-type="math/mathml"> <m

Abstract In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.